Welcome to Taylor Expansion! A Taylor expansion is a powerful mathematical tool that represents any smooth function as an infinite sum of polynomial terms. These terms are calculated using the function's derivatives at a single point, called the center point.
Let's walk through the step-by-step process of finding a Taylor expansion. First, we choose our function and center point. Then we calculate the function value at that point. Next, we find the derivatives and evaluate them at the center. Finally, we substitute everything into the Taylor formula. Here's an example with e to the x centered at zero.
Now let's see how Taylor polynomials visually approximate the original function. We start with a linear approximation using just the first two terms. Then we add more terms to get quadratic and cubic approximations. Notice how each additional term makes the approximation more accurate near the center point at zero.
Here are some of the most important Taylor series that you'll encounter frequently. The exponential function, sine and cosine functions, natural logarithm, and the binomial series. Each has its own unique pattern and convergence properties. These series are fundamental in calculus, physics, and engineering applications.
To summarize what we've learned about Taylor expansions: They represent functions as infinite polynomial sums calculated from derivatives at a center point. More terms give better approximations. Common series like exponential and trigonometric functions have standard forms. These expansions are essential tools in calculus, physics, and engineering for function approximation and mathematical modeling.